document.write( "Question 877860: Solve the following quadratic equation using factoring:
\n" ); document.write( "28x^2 + 30x + 32 = 0
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Algebra.Com's Answer #529611 by richwmiller(17219) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"28%2Ax%5E2%2B30%2Ax%2B32\"$ Start with the given expression.

$\"2%2814x%5E2%2B15x%2B16%29\"$ Factor out the GCF $\"2\"$ .

Now let's try to factor the inner expression $\"14x%5E2%2B15x%2B16\"$

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Looking at the expression $\"14x%5E2%2B15x%2B16\"$ , we can see that the first coefficient is $\"14\"$ , the second coefficient is $\"15\"$ , and the last term is $\"16\"$ .

Now multiply the first coefficient $\"14\"$ by the last term $\"16\"$ to get $\"%2814%29%2816%29=224\"$ .

Now the question is: what two whole numbers multiply to $\"224\"$ (the previous product) and add to the second coefficient $\"15\"$ ?

To find these two numbers, we need to list all of the factors of $\"224\"$ (the previous product).

Factors of $\"224\"$ :

1,2,4,7,8,14,16,28,32,56,112,224

-1,-2,-4,-7,-8,-14,-16,-28,-32,-56,-112,-224

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"224\"$ .

1*224 = 224
2*112 = 224
4*56 = 224
7*32 = 224
8*28 = 224
14*16 = 224
(-1)*(-224) = 224
(-2)*(-112) = 224
(-4)*(-56) = 224
(-7)*(-32) = 224
(-8)*(-28) = 224
(-14)*(-16) = 224

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"15\"$ :

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First Number	Second Number	Sum
1	224	1+224=225
2	112	2+112=114
4	56	4+56=60
7	32	7+32=39
8	28	8+28=36
14	16	14+16=30
-1	-224	-1+(-224)=-225
-2	-112	-2+(-112)=-114
-4	-56	-4+(-56)=-60
-7	-32	-7+(-32)=-39
-8	-28	-8+(-28)=-36
-14	-16	-14+(-16)=-30

From the table, we can see that there are no pairs of numbers which add to $\"15\"$ . So $\"14x%5E2%2B15x%2B16\"$ cannot be factored.

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Answer:

So $\"28%2Ax%5E2%2B30%2Ax%2B32\"$ simply factors to $\"2%2814x%5E2%2B15x%2B16%29\"$

In other words, $\"28%2Ax%5E2%2B30%2Ax%2B32=2%2814x%5E2%2B15x%2B16%29\"$ .

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